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.TH DSYTD2 l "15 June 2000" "LAPACK version 3.0" ")"
.SH NAME
DSYTD2 - reduce a real symmetric matrix A to symmetric tridiagonal form T by an orthogonal similarity transformation
.SH SYNOPSIS
.TP 19
SUBROUTINE DSYTD2(
UPLO, N, A, LDA, D, E, TAU, INFO )
.TP 19
.ti +4
CHARACTER
UPLO
.TP 19
.ti +4
INTEGER
INFO, LDA, N
.TP 19
.ti +4
DOUBLE
PRECISION A( LDA, * ), D( * ), E( * ), TAU( * )
.SH PURPOSE
DSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal form T by an orthogonal similarity transformation: Q' * A * Q = T.
.SH ARGUMENTS
.TP 8
UPLO (input) CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
.br
= 'U': Upper triangular
.br
= 'L': Lower triangular
.TP 8
N (input) INTEGER
The order of the matrix A. N >= 0.
.TP 8
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = 'U', the leading
n-by-n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading n-by-n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, if UPLO = 'U', the diagonal and first superdiagonal
of A are overwritten by the corresponding elements of the
tridiagonal matrix T, and the elements above the first
superdiagonal, with the array TAU, represent the orthogonal
matrix Q as a product of elementary reflectors; if UPLO
= 'L', the diagonal and first subdiagonal of A are over-
written by the corresponding elements of the tridiagonal
matrix T, and the elements below the first subdiagonal, with
the array TAU, represent the orthogonal matrix Q as a product
of elementary reflectors. See Further Details.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
.TP 8
D (output) DOUBLE PRECISION array, dimension (N)
The diagonal elements of the tridiagonal matrix T:
D(i) = A(i,i).
.TP 8
E (output) DOUBLE PRECISION array, dimension (N-1)
The off-diagonal elements of the tridiagonal matrix T:
E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
.TP 8
TAU (output) DOUBLE PRECISION array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
Details).
.TP 8
INFO (output) INTEGER
= 0: successful exit
.br
< 0: if INFO = -i, the i-th argument had an illegal value.
.SH FURTHER DETAILS
If UPLO = 'U', the matrix Q is represented as a product of elementary
reflectors
.br
Q = H(n-1) . . . H(2) H(1).
.br
Each H(i) has the form
.br
H(i) = I - tau * v * v'
.br
where tau is a real scalar, and v is a real vector with
.br
v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
.br
A(1:i-1,i+1), and tau in TAU(i).
.br
If UPLO = 'L', the matrix Q is represented as a product of elementary
reflectors
.br
Q = H(1) H(2) . . . H(n-1).
.br
Each H(i) has the form
.br
H(i) = I - tau * v * v'
.br
where tau is a real scalar, and v is a real vector with
.br
v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
and tau in TAU(i).
.br
The contents of A on exit are illustrated by the following examples
with n = 5:
.br
if UPLO = 'U': if UPLO = 'L':
.br
( d e v2 v3 v4 ) ( d )
( d e v3 v4 ) ( e d )
( d e v4 ) ( v1 e d )
( d e ) ( v1 v2 e d )
( d ) ( v1 v2 v3 e d )
where d and e denote diagonal and off-diagonal elements of T, and vi
denotes an element of the vector defining H(i).
.br
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